Motivation, resources

and the organization of the school system∗

Facundo Albornoz†,

Samuel Berlinski‡,

Antonio Cabrales§

April 2015

Abstract

We study a model where student effort and talent interact with parental and teach-

ers’ investments, as well as with school system resources. The model is rich, yet

sufficiently stylized to provide novel implications. We can show, for example, that an

improvement in parental outside options will reduce parental and school effort, which

are partially compensated through school resources. In this way we provide a rationale

for the existing ambiguous empirical evidence on the effect of school resources. We

also provide a novel microfoundation for peer effects, with empirical implications on

welfare and on preferences for sorting across schools.

JEL-Classification: I20, I21, I28, J24

Key-words: education, incentives, school resources, parental involvement, school sort-

ing, peer effects.

∗We thank Andres Erosa, Esther Hauk, Elena Inarra, Gilat Levy, Diego Puga, Miguel Urquiola, andseminar participants at University of Birmingham, Columbia University, LACEA, IMDEA, XIII UrrutiaElejalde Summer School on Economics and Philosophy, Science Po, Seoul National University, StockholmSchool of Economics and Warwick. Albornoz is grateful for support from the ESRC (RES-062-23-1360),Cabrales acknowledges the support the Spanish Ministry of Science and Technology under grant ECO2012-34581.The views expressed herein are those of the authors and should not be attributed to the Inter-AmericanDevelopment Bank.†falbornoz@udesa.edu.ar‡samuelb@iadb.org§a.cabrales@ucl.ac.uk

1

1 Introduction

Education policy is at the forefront of the social and political debate. The belief that edu-

cation is a catalyst for a better and more equitable society ensures its role in the political

agenda, in both developed and developing countries. As a consequence, a variety of policies

and reforms are continuously being proposed with the objective of improving the outcomes of

the education system. Surprisingly, the implementation and evaluation of these policies often

overlooks the changes in behavior they can induce in the actors involved in the education

process. For example, the debate about the role of education resources on student learning

does not usually take into account behavioral responses from parents and school adminis-

trators. Similarly, proposals of school vouchers generally disregard how different ways to

sort students into schools would affect the determination of school policies, or their influence

on parental involvement in education and, crucially, the political support for such schemes.

In this paper, we study a model of education where student learning effort and outcomes,

parental and school behavior, and public resources devoted to education are endogenously

determined in an integrated theoretical framework.

In our model, the determination of educational outcomes is a process involving five par-

ticipants: children, parents, headmasters, teachers and the policymaker. Each child chooses

a certain level of effort devoted to learning. More able children obtain a higher learning out-

come from a unit of effort. Altruistic parents and schools affect the effort decision through

motivation schemes. However, inducing effort is costly for parents as well as for schools.

Both for parents and teachers, there is an opportunity cost for the time involved in setting

up and executing the motivation plan, which may also include monitoring or helping children

with their learning tasks (such as homework). How costly it is for schools depends on their

resources (class sizes, for example), which are determined by the policymaker, as well as the

talent of their teachers. This integrated framework provides an accurate description of the

workings of the education process: parents, students and the education system interact in the

determination of school resources, education quality, school and parent education methods

and, through all these, on students results. An advantage of our framework is its tractability,

which allows us to analyze many important dimensions of the education process.

We start with a case where children are homogeneous in terms of innate ability and

parents’ characteristics, such as talent and opportunity cost of time. We find that the

strength of parental and school involvement and resources devoted to education increase

with student innate ability. The results are less clear cut when we analyze the impact of

an increase in the opportunity cost of time associated with parental involvement in the

2

learning process of their children. This introduces, in a natural way, the connection between

labor market conditions and the direct involvement of parents in the education of their

children. This link goes beyond the hourly wage. For example, it can also capture changes

in opportunities and incentives for female participation in the labor market. In either case,

the strength of parental involvement is decreasing in the opportunity cost of time.

The interaction between the school system and parental inputs is the reason why political

considerations are important. As the parental opportunity cost of time increases, they would

like to rely more heavily on schools for motivating their children, which triggers actions by

those responsible for the education system. The policymaker anticipates participant choices

and satisfies parental wishes by increasing the resources devoted to education. Interestingly,

the increase in school resources may not be accompanied by an overall increase in educational

attainment. A result far too familiar for those in the education policy arena.1 Our model

can predict “disappointingly” weak effects of school resources on student results even in

situations where school resources do in fact affect learning cæteris paribus . The weak effect

can be rationalized because cæteris paribus does not hold when resources increase. Parental

involvement decreases because of a change in their opportunity costs. School resources

increase to compensate for this reduction.2 These additional resources have in fact an effect,

but this is not apparent because of concomitant changes in parental involvement in the

education process. This process could also explain why the increase in expenditures per

student observed in many countries during the last decades has not been followed by better

test scores or improvements in other measures of student performance.3

We then allow for children to differ in terms of ability and parental opportunity costs of

time, which leads to a number of insights. First, as the school determines their motivational

1The empirical findings of the class-size literature are ambiguous. For example, Angrist and Lavy (1999);Krueger (1999); Urquiola (2006) report positive results of class-size on student attainment while others(Hanushek (2003); Hoxby (2000); Leuven, Oosterbeek, and Rooonning (2008); Anghel and Cabrales (2010))find no gains.

2As an early example of the connection between parents’ opportunity costs and school resources, Flyerand Rosen (1997) attribute the increase in school expenditures taking place in the United States during1960-1990 to the growth in female labor-force participation.

3See Hanushek (1998) for the case of the US. This result also provides a possible reason for a low cross-country correlation between education expenditures and school attainment levels results in standardizedtests. See for example, Hanushek (2006). And this “anomaly” has been recognized for a long time. Forexample, in words of The Economist, “Glance at any league table of educational performance and you willfind several Asian countries bunched near the top. The achievements of the region are a puzzle to peoplewho think that educational success is all a matter of expenditures. Even in Japan most of the schools areill-equipped by comparison withe their western equivalents [...] The children are driven on by intense familypressure. Parents badger their children to succeed, but they also make big financial and personal sacrificesto help them do so. Mothers help their children with their homework [...] Fathers promise fancy toys andactivities in return to examination success...” Quote from The Economist, November 21st 1992.

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policy for the average individual in the classroom, school involvement is positively affected

by the mean ability of their students. In equilibrium, this affects the intensity of parental

involvement. Thus, peer effects arise endogenously, as the choice of effort and motivational

policy in equilibrium depend on the average ability of the student in the classroom. This

effect is reinforced by the determination of school resources. The policy maker decides the

level of resources optimally given the characteristics of the school attended by the median

voter’s child. Thus, the decision on school resources will be based on the average ability

of this school and on the median child’s ability. As a consequence, student effort depends

on the mean abilities of peers at her/his school, plus the ability of the median child and

his peers. Our model generates in this way a microfoundation for peer effects, rather than

assuming they come from some exogenous “contagion” process, as it is more common in the

literature.

This matters because the existence of peer effects has been the subject of a recent con-

troversy.4 What is important for us about this literature is that, as our paper suggests,

the existence or not of peer effects seems to depend on the interaction between the peers

and school organization. For example Abdulkadirolu, Angrist, Dynarski, Kane, and Pathak

(2011) show that pilot schools, which operate like public schools but are more selective pro-

duce no measurable outcomes, whereas charter schools, which change organization at the

same time that students are more strongly selected do present some changes.5

In this context, an increase in the opportunity cost of the median parent raises similar

issues to those identified for the homogeneous case. However, the link between median

child characteristics and individual effort generates a channel through which changes in

the distribution of income (or talent) can affect the educational choices of households and

schools. For example, an increase in the income of the median child’s household will generate

an increase of resources in the system. This will induce a positive effect on households in

lower parts of the income distribution even if their incomes do not change. And the other

way around, if the income of the median does not change (or it changes very little) in an

environment where mean income is increasing markedly, there will be few changes in school

outcomes (or even a regression) at a time when income appears to be fast increasing.

We finally incorporate private schools in our setting. We show first that a mixed education

system with public and private schools generates endogenous sorting by either parental

4Some papers (such as Imberman, Kugler, and Sacerdote (2012), using data from a natural experiment)find strong evidence of their presence, whereas other papers do not (Abdulkadiroglu, Angrist, and Pathak(2014) and Dobbie and Jr. (2011)), using evidence from lotteries or entry exams into selective school).

5Also, Duflo, Dupas, and Kremer (2011) find that strong peers have an effect only in some kind of schools,those that are non-tracking.

4

opportunity cost of time or student talent. That private schools imply stratification is a

well established insight of the literature. For example, Epple and Romano (1998) show that

the existence of positive peer-effects induces schools to be perfectly ordered according to

peer-group ability and may lead to stratification by income. Our model also displays an

assignment equilibrium with top-down sorting.6 Yet, the distinctive feature of our model

is the focus on the endogenous determination of peer-effects, and hence school quality and

policy choices, as a result of the interaction of parents and the school system. This allows us

to analyze in detail the effect of policies increasing school choice for parents, like a voucher

scheme. We show that, even if the median voter is favored (and hence the voucher policy

approved), the reaction of schools to the changes in classroom composition, will increase

inequality in student achievement. This is because the worsening of peer-effects in the

schools where the less able students stay (the “cream-skimming effect”) is magnified by

the responses of other actors. The headmasters will decrease motivational strength at those

schools, and policymakers will decrease the resources devoted to them. Hence, our framework

allows us to understand in a simple way the effects of students’ quality, and the reaction of

other actors to this quality, on the incentives for school sorting.7 Our result that sorting

feeds back on school quality and classroom peer-effects has important implications for the

analysis of school choice programs. For example, Altonji, Huang, and Taber (2010) find that

the cream-skimming effect on high school graduation was modest. However, their estimates

did not consider how parents, teachers and headmasters would react to different classroom

compositions. Our contribution is to show that these reactions should be incorporated in

the empirical literature.8

Stratification by income may also emerge if the labor market considered school reputation

as a signal of students’ skills. MacLeod and Urquiola (2009) show that this is indeed the

case. In their model, individual skills are partially observed by standardized tests and

the reputation of the school attended by each student. As reputation depends on classroom

6We can also obtain stratification by income by simply assuming a positive correlation between abilityand income.

7Previous papers focus on just one of these elements. While Urquiola and Verhoogen (2009) and Epple,Romano, and Sieg (2006) focus on educational quality, Epple and Romano (1998, 2008) and Epple, Figlio,and Romano (2004) concentrate purely on peer-effects.

8Teachers also react to changes in classroom composition. For example, teachers may be more efficientin fairly homogenous classrooms (Duflo, Dupas, and Kremer, 2011) or with students of the same race (Dee,2004). They may also react to the educational goals established through the implementation of new educationpolicies. For example, Aucejo (2011) find that teachers in North Caroline have refocused their teaching effortsto relatively low performing students after the implementation of the No Child Left Behind program. Ourmodel summarizes these concerns in a single statistic capturing the strength of motivation at the school,which we endogenously derive.

5

composition, school reputation reflects the quality of peers. As the reputation of the attended

school increases expected wages, school quality would induce lower individual effort. In

our model, the effect of peers on individual effort is the opposite: better peers encourage

schools to provide incentives and induce higher student effort. While the empirical literature

has not yet established which one prevails, we see both effects as complements: school

motivation schemes are arguably more relevant in early and primary education while the

school reputation effect on student effort may be more prevalent for older students.

The relevance of behavioral responses to education policy changes is attracting growing

interest in the empirical literature. Das, Dercon, Habyarimana, Krishnan, Muralidharan, and

Sundararaman (2013) emphasize behavioral aspects associated with higher school inputs. In

particular, as school and household educational spending are substitutes, an unanticipated

increase in school funding reduces parental school expenditures. Pop-Eleches and Urquiola

(2013) uncover the effect of access in Romania to better (high) schools on student outcomes

and parental behavior. They identify, for example, the positive effect of school quality on

students’ achievement at the national standardized examinations. Importantly for our paper,

they find that parental effort and quality-improving school activities are substitutes for each

other. Additional evidence of the substitution between parental effort and school resources

is provided by Houtenville and Smith Conway (2008).

Stinebrickner and Stinebrickner (2008) and De Fraja, Oliveira, and Zanchi (2010) pro-

vide empirical evidence on the positive impact of parental and student effort on educational

achievement. De Fraja, Oliveira, and Zanchi (2010) adds the dimension of school effort

which is positively associated with examination scores. They also provide tentative evidence

of feedback effects between parental, student and school efforts. They find, for example, that

parental involvement induces more effort from their children which leads to higher involve-

ment from schools.9 In real life, parental involvement may take different forms. Doepke and

Zilibotti (2014) study a model where different parenting styles (authoritarian, permissive

and authoritative) are endogenously determined by the interaction of parental preferences

and the socioeconomic environment. Although we do not distinguish between forms of par-

enting, our approach is complementary insofar as our endogenous determination of parental

involvement intensity.

Our work is also related to the literature that studies the endogenous determination of

class size. In Lazear (2001) class size is decided by schools according to student behavior.

9Sahin (2004) provides another example of how parent and student responses affect the impact of educa-tion policies for the case of higher education tuition subsidies. Evidence of the interaction between parentsand the school system mediated by monitoring of schools is offered by Liang and Ferreyra (2012).

6

For example, when students have a shorter attention span (i.e., they can be distracted more

easily) students should be sorted in smaller classrooms as they require closer attention. In

Urquiola and Verhoogen (2009), schools differ in productivity and offer different quality levels

(school size). As parents differ in earnings, sorting between schools with different class sizes

arises naturally. Our model offers a complementary mechanism behind the determination

of class size, which relies upon the interaction of parental and school motivation, which is

partly determined by the government through the (strategic) choice of school resources.

We organize the paper as follows: In section 2, we set up the model. We characterize the

equilibrium in section 3, where we discuss the interdependence between parental and school

motivation systems, school resources and student performance. Section 4 analyzes a case

with private schools and endogenous segregation. We conclude in section 5.

2 The model

Our model has four participants: children, parents, headmasters, and the policymaker.10

There are different ways to specify the production function of human capital (H). As a

general level, we assume that learning requires innate talent (υ) and effort (e). Parents and

schools can affect this process through different interventions and actions. It is not clear

however whether parental and school involvement induce learning through a direct impact

in H or via inducing changes in effort. In the core of the paper, we emphasize that parental

and school interventions affect effort and therefore learning in a indirect way. We leave

for the Appendix A.1 the discussion of actions that directly impact in the accumulation of

human capital such as parents hiring private tuition for their children.

Assuming that parents and schools can affect the learning effort is consistent with what

psychologists call “Achievement Goal Theory” (see, for example, Covington (2000)). Accord-

ing to this perspective, achievement goals influence the quality, timing and appropriateness

of the students’ engagement in their own learning (e.g., analyzing the demands of school

tasks, planning and allocating resources to meet these demands, etc.). This effort together

with innate ability affect the student’s accomplishments. In our model, parents and teachers

play a key role in influencing the student’s achievement goals and, in turn, their effort.

Two kinds of goals have been predominantly studied in achievement goal theory: learning

goals and performance goals. Learning goals refer to increasing one’s competency, under-

standing and appreciation of what is learned. Performance goals involve outperforming

10For simplicity, we refer to students as she and to teachers as he.

7

others in tests or other achievement measures. In our model, parents and teachers can be

thought as focusing on affecting learning goals.

Crucially, inducing effort is costly for parents as well as for schools. Both for parents and

teachers, the main cost is the opportunity cost of the time involved encouraging students in

the pursuit of learning goals. The cost for schools depends as well on the level of resources

(for example, class size), which are determined by a policymaker.

Student performance and children’s short-term utility Human capital for child

i, Hi, (measured in monetary terms) is a linear function of her effort, ei, and a productivity

parameter, υi. In particular, we assume,

Hi = υiei. (1)

In the core of the paper, we assume υi to be exogenous, which can therefore be interpreted

as a measure of innate talent. We discuss the implications of different specifications of Hi in

Appendices A.1, A.2 and A.3.

Children internalize only imperfectly the effect of their effort on human capital. The

role of parents and teachers is to induce students to exert costly effort through different

motivational tools. In particular, we postulate that children’s short-term utility is given by:

USi = c1jei + c2jei −1

2e2i . (2)

Exerting effort implies a cost that takes a quadratic form. Parent and teacher’s involvement

enters positively in the utility of the child, c1jei + c2jei, with the sub-index j denoting the

school attended by the child i. c1j (c2j) is a summary of the strength of parental (teacher)

incentives for every unit of child’s effort. Therefore, c1jei + c2jei is the child’s total reward

for her effort.11

We are agnostic about the real life characterization of c1j (c2j). We think about it as

a reduced form of a complex incentive structure that arises within families (schools). For

example, rewards can be extrinsic such as buying presents or taking a child to the park, but

c1j may also capture intrinsic forms of motivation that require direct parental involvement

11This linear structure of rewards is analogous to the linear contracts that are widespread in reality.Bose, Pal, and Sappington (2011) justify the use of linear contracts by showing that, in a broad class ofenvironments, the optimal linear contract always secures for the principal at least 90% of the expected profitsecured under a fully optimal contract. Furthermore, Carroll (2013) considers an environment in which theprincipal is uncertain about what the agent can and cannot do and wants to write a contract that is robustto this uncertainty. He finds that, under very general circumstances, the optimal contract is linear.

8

in the learning process and that affect the value of acquiring knowledge. In other words, the

role of parents (teachers) is to help children intenalize the value of ei (and ultimately Hi) in

their utility function.

Pomerantz, Moorman, and Litwack (2007) argue that parental involvement12 in school

can enhance children’s school performance through either skill development (e.g., by instruct-

ing children) or motivational development (e.g., by providing intrinsic reasons for learning).

Some motivational activities clearly play both roles. Take for example, helping with home-

work. On the one hand, it increases the productivity of the children effort (i.e., is tantamount

to an increase in υi). On the other hand, it increases the intrinsic value of studying for the

child because parental involvement and concerns help children internalize the benefits of their

effort. As discussed above, we focus on the motivational channel and leave for the appendix

the discussion of parental interventions that directly affects productivity of learning effort

(Appendix A.1).

The additive specification for parent and teacher incentives implies that parents and

school involvement are substitutes.13 This assumption is supported by the empirical evidence

suggesting that parental and school involvement are substitutes (Houtenville and Smith

Conway, 2008; Pop-Eleches and Urquiola, 2013), but carries no qualitative implications as

we discuss below and in Appendix A.3.

Of course, effort may also be obtained through negative reinforcement (punishment). In

this case, we could have written the utility in the alternative way:

USi = −cij (1− ei)−1

2e2i = cijei − cij −

1

2e2i .

As will be clear below, this utility induces the same optimal action from her as the one

we examine. Thus, provided the costs of the two incentive systems can be written in the

same way, there will be no difference in any equilibrium value.14

12Following Hoover-Dempsey and Sandler (1997), parental involvement includes home-based activitiesrelated to children’s learning (e.g., helping with homework, discussing school events) and school-based in-volvement (e.g., volunteering at school, attending school functions).

13This formulation also assumes that effort is perfectly correlated with human capital. Nothing majorwould change in the model if the relationship between effort and output were noisy, provided effort wereobservable and contractible. We do not think this is an unrealistic assumption in the case of children. Non-observability of effort with noisy output would be, of course, more complicated but we do not think any newinsights would be gained by studying that case.

14In our setting, this is the case because we assume that parents care about H but not about her shortterm utility.

9

The parents’ utility We assume that every parent has one child and that their utility

is influenced by the sum of her performance and their own welfare, denoted by Wi (also

measured in monetary terms, as Hi).15 Hence,

UPi = Hi +Wi.

Parental welfare depends on the time spent at work or pursuing leisure activities. This

is the total time (T ) available minus the time spent with the child as a consequence of the

reward scheme (ti). The effective reward (c1iei) depends on time (ti) and parental efficiency

(υPi) of parent i to generate a given effective reward, so that c1iei = 2tiυPi . The parameter

υPi captures the fact that some parents are better at motivating than others and consequently

they generate a larger reward for any given amount of time devoted to their children. That

is, we associate productivity of parental involvement with some innate (exogenous) attribute,

such as talent or persuasiveness. Implicitly, we are imposing that the time devoted inducing

effort, for a given effective reward, is decreasing in the ability of the parent to generate it. In

Appendix A.4, we study the effect of parental attention, where υPi is derived endogenously

and shown to depend on the opportunity cost of time.

Thus, letting ψi be the opportunity cost of parent i yields,

Wi = (T − ti)ψi =

(T − c1iei

2υPi

)ψi,

and, therefore,

UPi = υiei +

(T − c1iei

2υPi

)ψi. (3)

Since ψi is an opportunity cost of time for parents, it can be interpreted as a wage rate,

although it can also be the value of leisure or something else.16 Hence, in the remainder of

the paper we often refer to this parameter as parental income.17

15The utility function below does not internalize the child’s cost of effort, so it is not purely “sympathetic”.On the one hand this is reasonable, since this cost of effort is not observable. But we have also done thecomputations with strictly sympathetic parents’ utility and there are no significant changes.

16In our context, the marginal utility of money earned by parents is linear. Hence, the value of time forparents with high wages is larger. Things may be different with concave utility for money. In that case, lowwage earners may have a higher opportunity cost of time. We are agnostic about which parents have thehigher opportunity cost of time in reality.

17Notice we are not considering parental teaching time in this specification of parents’ welfare. How-ever, this would be easily incorporated into the analysis by assuming Hi = Ti + Wi, and Wi =(T − c1iei

2υPi− 1

2T 2i

υPi

)ψi where t′i = T 2

i /2υPi represents the actual units of time a parent devotes to teach-

ing and Ti represents effective teaching time. Notice as well that this component is independent of student

10

Remark 1 The parents also value (negatively) the taxes that the government will need to

levy in order to pay for school resources. We do not include them here explicitly in order to

avoid an excess of notation. Given the quasi-linearity in income of utility and that taxation

is already decided at the time parents choose their effort, the amount of those taxes do not

affect the parental effort decision.

The school objective function According to Covington (2000), every classroom re-

flects rules that determine the basis on which students will be evaluated and how rewards

will be distributed. We assume that teachers reward students based on individual learning

expectations and not in a competition for one or a few prizes. The school rewards (and

therefore the learning goals) are determined by the teachers’ effort.18

We consider first the case of public (state) schools. In this case, we assume that the head-

master chooses the intensity of motivation (summarized in the parameter c2j) and therefore,

teachers’ effort, in order to maximize the sum of the average student performance and the

welfare of the average teacher in the school. Of course, a headmaster might have different

goals and statistics to measure school’s performance. In Appendices A.5 and A.6, we change

the goal of maximizing average student performance in two crucial ways. In Appendix A.5,

we study a case in which the headmaster cares more about a group of students according

to their place in the distribution of human capital.19. We also study the case where head-

masters are imposed an exogenous standards (Appendix A.6), such as, for example, that the

average score be higher than a particular threshold (reference). In both cases, we show that

the main messages of our analysis are robust to these variations in the headmaster’s goals.

The average welfare of a teacher is determined by the difference between the total time

available to him and the average time he devotes to his students, which we assume is a linear

function of c2jei, and inversely related to υTj , the motivational ability of a representative

teacher at school j. Thus, letting γ be the opportunity cost of teachers’ time,20

effort and has no interaction with c1i and therefore it will carry no effect on our results.18For simplicity we assume that the costs of creating the rewards for the teachers are unrelated to student

ability.19More specifically, we study an alternative formulation where the headmaster cares about a weighted

average of the mean and any order statistic of the distribution of human capital within the school. SeeAucejo (2011) for an example of how teachers adjust their population targets within a classroom to changesin the goals imposed by the authorities.

20We assume that the opportunity cost of the teacher γ is unrelated to her talent υTj . This is done tosimplify notation. Little is changed if γ depends on υTj

. It is worth pointing out that this is not the teacher’swage but rather the value of his alternative use of time which can either be leisure or the compensation hewill get from, say, private tutoring.

11

UHMj =1

Nj

∑i∈j

υiei +

(T − nj

Nj

∑i∈j

c2jei2υTj

)γ, (4)

where Nj is the total number of students in school j and nj is the number of students per

classroom.

The policymaker objective function The policymaker maximizes the complete util-

ity of the (median-voter) parent (denoted by Pi) which, as discussed in remark 1, requires

adding the cost of school resources (1/n). The decision about resources is taken and an-

nounced before parents and headmasters simultaneously decide their actions (c1i and c2j).

Therefore, the cost of resources does not appear in UPi or UHMj because parents and head-

masters take them as given when making decisions about their involvement. These costs are

paid by parents through general taxation, which parents care about, and are internalized by

the policymaker when deciding n.

The cost depends on the number of classes to be manned. That is, the ratio of total

number of students in the system, N , to the number of students per class, n. For simplicity,

we assume that all public schools operate at full capacity and with the same class size so that

nj = n for all j. Manning costs are assumed to be quadratic in the number of classrooms

N/n. This can be justified by taking into consideration that the state has monopsony power

in the market for teachers and faces a marginal cost function that increases in the number of

teachers hired. This is so, for example, because to attract one more teacher the monopsonist

has to pay an extra cost, since the marginal potential teacher needs a higher reward to be

attracted to the profession.

Thus, we can represent the policymaker’s preferences as,

UPM = UPi −1

N

ω′

2

(N

n

)2

, (5)

where ω′ is a constant parameter summarizing the cost of the chosen class size and 1N

ω′

2

(Nn

)2

is the per capita cost of that class size.21 Our formulation assumes that schools are financed

out of lump sum taxation and the government keeps a balanced budget. For ease of notation,

21Alternatively, the formulation can be reinterpreted by assuming that the average quality of teacherswhen hiring k people is 1/

√k. Hence, in order to man N/n classrooms and keep the quality of teaching per

classroom constant,(N/n)2

teachers need to be hired. If ω′/2 is the wage per employed person the total cost

of N/n classrooms is ω′

2

(Nn

)2and the cost per student 1

Nω′

2

(Nn

)2.

12

in the remainder we denote ω = ω′N, so that

UPM = UPi −ω

2

1

n2, (6)

The structure of the game Summarizing, the policymaker announces first the policy

variable (n). After this announcement, parents and headmasters simultaneously decide their

optimal levels of rewards per unit of effort c1i and c2j, respectively. After observing parents’

and schools’ announcements, the children decide their optimal level of effort, ei.

3 Equilibrium

We solve the game by backward induction.

3.1 Students’, parents’, and school choices

From equation (2), it follows that the optimal student action is

ei = c1i + c2j. (7)

Substituting this expression into the parents’ utility, equation (3), we obtain

UPi = (c1i + c2j) υi +

(T − 1

2υPic1i (c1i + c2j)

)ψi

The first-order conditions for the parents’s problem is then

υi −(c1i +

c2j

2

) ψiυPi

= 0.

Given that this condition is sufficient, the optimal choice of the parent is

c1i = max

υPiυiψi− c2j

2, 0

(8)

which is always non-negative given that motivating requires investing parental time.

It is clear from the expression for c1i that the strength of parental involvement is in-

creasing in the abilities of the child (υi) and the parent (υPi), and decreasing in the parental

opportunity cost of time (ψi). Also, equation (8) shows the negative relationship between

c1i and c2j. When motivation in the school is high, the gains from additional effort induced

13

by parental motivation are smaller. We shall discuss below how both incentive schemes may

compensate each other in responding to changes in ψi, υPi and υi.

At this point, the following clarification is in order:

Remark 2 The assumption of substitute rewards is not essential for the negative rela-

tionship between c1i and c2j. A similar result is obtained with other specifications where

parental and school efforts are complements. For example, when Hi = υieiα, with α < 1

and cij = c1ic2j (see Appendix A.3). The driving force in our result is that greater school

incentives will reduce the marginal benefit of parental effort.

By substituting the optimal choice of children’s effort into the utility function of the

headmaster (4) we obtain:

UHMj =1

Nj

∑i∈j

(c1i + c2j) υi +

(T − nj

Nj

∑i∈j

c2j (c1i + c2j)

2υTj

)γ.

It follows that an interior solution for the headmaster’s optimization problem implies

c2j =υTjυj

γnj− c1j

2, (9)

where υj is the mean student ability and c1j is the mean parental reward for the students

attending school j.

We can interpret υTjυj as the school quality. Clearly, talent of teachers plays a key role

in defining school’s quality. Also, as in Epple and Romano (1998), quality of schools depends

on the average of peer’s talent. A higher average peer talent is associated with greater class-

room motivation. This association, which will tend to amplify school differences in student

performance, will be important in the emergence of peer effects and will provide the moti-

vation for a segregated school system, as we shall discuss below. Also, raising opportunity

costs for teachers and larger class sizes lead to lower levels of classroom motivation. It is, of

course, through this latter channel that public resources affect student performance. Finally,

a high level of parental involvement is associated with lower school incentives.

3.2 Equilibrium values for c1, c2 and n

3.2.1 Homogeneous children

In order to solve for the first stage of the game, let us first assume that children, parents

and teachers are all identical, so that υi = υ, υPi = υP and ψi = ψ for all i ∈ 1, .., N and

14

υTj = υT for all j. In this case we have that c1i = c1 for all i and and c2j = c2 for all j.

Therefore, from (8) and (9), an interior solution for c1 and c2 implies

c1 =υPυ

ψ− c2

2(10)

and

c2 =υTυ

γn− c1

2. (11)

In the first stage, the policymaker considers the optimal level of school and parental

involvement. After substituting (11) in (10) and plugging the resulting expression, together

with (7), into (6) we obtain

UPM =2υ2

3

(υPψ

+υTγn

)+

(T − 2υ2

9υP

(2υ2

P

ψ2+υPυTψγn

− υ2T

(γn)2

))ψ − ω

2

1

n2

As we show for the general heterogeneous case in the next section, preferences are uni-

modal in the policy parameter n and we can apply the median-voter theorem to obtain the

optimal level of n. The interior solution that results from maximizing the above expression

with respect to n is (assuming ω > (2υυT/3γ)2 (ψ/υP ))

n =ω −

(2υυT

3γ

)2ψυP

υTγ

(2υ3

)2 . (12)

Therefore, class size is increasing in the cost of manning classes, ω, parental motivation

ability υP and the opportunity cost of time of teachers, γ, and decreasing on student ability,

υ, teacher talent υT and parental opportunity cost of time, ψ. The post-war decades were

characterized by a large increase in female labor-force participation, arguably compatible

with an increase in ψ. This would suggest a decrease in n (probably moderated by a parallel

increase in γ). Indeed, Flyer and Rosen (1997) find a connection between the rapid growth

in public expenditures associated with elementary and secondary education in the US and

the rising value of women’s time. Between 1960 and 1990 the real costs of elementary and

secondary education increased by 300 percent. The main cause of the increase in expenditures

per student was rising school staff, with a halving of the number of the student-teacher

ratios between 1950 and 1990. The paper studies further the connection between female

labor force participation and student-teacher ratios using a panel of 50 US states during

4 decades. They found that increases in female labor force participation rates explain a

15

significant part of both the level and growth in states’ student-teacher ratios. Moreover,

the findings are robust to other plausible explanations such as increases in unionization and

changes in fertility patterns.

We substitute the optimal level of n in equation (11) and then into equations (7) and

(10) to obtain the equilibrium values of c1, c2 and e. These are:

c1 =2υPυ

3ψ

2ω − 3(

2υυT3γ

)2ψυP

ω −(

2υυT3γ

)2ψυP

,

c2 =2υPυ

3ψ

3(

2υυT3γ

)2ψυP− ω

ω −(

2υυT3γ

)2ψυP

,

and

e =

2υP υ3ψ

ω

ω −(

2υυT3γ

)2ψυP

. (13)

From inspecting the above expressions it becomes clear that a necessary condition for a

positive c1 is

ω >3

2

(2υυT3γ

)2ψ

υP. (14)

This condition is sufficient for n and e to be positive as well. A necessary condition for

a positive c2 is

ω < 3

(2υυT3γ

)2ψ

υP. (15)

The comparative static results of school and parental involvement with respect to υ and

υT are simple. An increase in student ability (υ, which one can think of as innate or the

result of early parental stimulation) leads to stronger school and parental motivation. The

same effect is associated with higher parental motivation ability. These factors in turn induce

higher student effort.

The effect on effort of an increase in ψ is ambiguous. First, a higher ψ imposes a higher

opportunity cost for parents to engage in motivational activities. Hence, c1 is decreasing

in ψ. The school system reacts to this by reducing n and therefore c2 is increasing in ψ.

16

The driving force for this result is that the policymaker devotes more resources to classroom

education, which lowers the cost of inducing effort by the school. Conversely, lower class size

and the consequent stronger school ethos, reduces the gain from staying at home inducing

children’s effort.

The optimal effort decision is equal to the sum of parental and school involvement. When

the opportunity costs for parents increases the resulting fall in parental involvement is not

always fully compensated by the school system. Therefore, the net effect of an increase of ψ

on student performance may be negative. To see this:

∂e

∂ψ= −2υPυω

ω − 2(

2υυT3γ

)2

ψ[ωψ −

(2υυT

3γ

)2ψ2

υP

]2

which implies that effort is decreasing in ψ when

ω > 2

(2υυT3γ

)2ψ

υP.

Notice that from (14) and (15)

3

(2υυT3γ

)2ψ

υP> ω >

3

2

(2υυT3γ

)2ψ

υP

so that effort and school performance can be both increasing or decreasing within our para-

metric range.

The same ambiguity, but in the reverse direction, is generated by differences in parental

motivational talent υP . This is the case since in all expressions we have ψ/υP .

The model highlights an important issue on empirical estimates of the effect of class-

size (or educational resources in general) on educational outcomes. Following Todd and

Wolpin (2003), the typical paper considers an educational production technology which is a

function of a set of school inputs, like class-size, and household inputs. Like in our model,

optimizing agents (see, for example, Glewwe (2002)) determine their input choice based on

a set of market and shadow prices. Isolating the effect of class size, or any other input, on

educational outcomes in empirical work is hard because it essentially requires conditioning

on all inputs. In general, what we can do is to find an exogenous source of variation in a

market or shadow price -excluded by definition from the production function- which affects

directly the input of interest and indirectly other inputs of the production function. Thus,

we estimate what is called in the jargon a policy impact which contemplates the direct effect

17

of the input of interest, say class-size, and changes in other inputs -behavioral changes. A

key contribution of our model is to highlight that different sources of variation for class-size

result in different behavioral responses and therefore different policy parameter estimates.

For example, our model has identified two sources of variation for class size, ω and ψ,

that may lead to different policy estimates of the impact of class size on student perfor-

mance. As we pointed out above, an increase in the opportunity cost of parents and a fall

in the cost of manning classes both lead to lower class sizes. However, a fall in the cost of

manning classes leads to a unambiguous increase in effort (as can be seen from (13)) and

an improvement in student performance. The fact that an increase in class size may lead to

different effects depending on the source of variability that generates these changes provides

an interesting lens through which we can interpret the findings in the empirical literature.

When the source of variability for class size comes from exogenous changes in the costs of

manning classes, like in most randomized experiments (see, for example, Krueger (1999)) we

should expect positive impacts on student performance. However, in cross-country or cross-

state panel studies, where differences in resources, like class-size, may be the consequence

of increases in opportunity costs for the median parents, we may find it more difficult to

observe improvement in educational performance.

3.2.2 Heterogeneous children

We relax now the assumption of identical children. For expositional clarity, we define the

following parameter:

Ωj ≡1

Nj

∑i∈j

υPiυiψi

.

In words, Ωj is the average at the school level of student ability times the ratio between

parental motivational ability and their opportunity cost of time. Thus, each school j is

associated with a particular Ωj.22

To obtain the utility of the policymaker, we substitute (9) in (8) and plug the resulting

expression into (7). This yields

e∗i =υPiυiψi

+1

3

(2

γnυTjυj − Ωj

), (16)

where υj is the average student ability in school j. Thus, equation (6) becomes:

22Notice that in the plausible cases where υi, υPi and ψi are correlated, the ranking of schools would beinvariant to whether the ranking is based on υ, υP , ψ or Ω (clearly, in different directions for each one).

18

UPM =

(υPiMυiMψiM

+1

3

(2

γnυTjMυjM − ΩjM

))υiM +

+

(T − 1

2υPiM

((υPiMυiMψiM

)2

− 1

9

(2

γnυTjMυjM − ΩjM

)2))

ψiM −ω

2n2,

where M stands for the median voter and, therefore, υjM and ΩjM express the characteristics

in the school the median voter attends.

The following lemma states that preferences are unimodal in the policy parameter, n,

and proves the validity of the median-voter theorem to determine school resources (n) in our

framework.

Lemma 1 Preferences of parents with respect to n are unimodal for all parents if 1Nj

∑i∈j

υPiυiψi

<

3 mini∈1,...,NυPiυiψi

.

Proof Please see Appendix B.

The sufficient condition for this result means that the minimal value of υPiυi/ψi (the ratio

of parental teaching talent to her opportunity cost, times the child’s talent) is not lower than

a third of the average value of υPiυi/ψi in the population.

The first-order condition for the policymaker’s maximization problem, provided an inte-

rior solution exists, is:

∂UPM∂n

= −2

3

υiMυTjMυjMγn2

− 2

9

ψiMυTjMυjMγυPiMn

2

(2

γnυTjMυjM − ΩjM

)+ω

n3= 0. (17)

We can make the following assumption to reduce notational complexity:

Assumption 1υPiM υiMψiM

= ΩjM .

This means that the ratio of parental and individual parameters for the median child is

equal to the average ratio in her school. Using this assumption, equation (17) simplifies to:

∂UPM∂n

= −2

3

υiMυTjMυjMγn2

+2

9

υTjMυjMυiMγn2

−(

2υTjMυjM3γ

)2ψiM

υPiMn3

+ω

n3= 0

19

and thus we obtain:

n =ω −

(2υTjM υjM

3γ

)2 ψiMυPiM

49

υiM υTjM υjMγ

. (18)

A positive class size requires ω − 49γ2jmM

(υjM )2 ψiMυPi

> 0. Like in the homogeneous case,

class size increases with the opportunity cost of manning classes. School resources are in-

creasing in the opportunity cost of the median parent and the ability of the median child as

well as the quality, υTjMυjM , of the school she attends.

From the derivation of (18) it is clear that parents of children with υi above υiM would

like the level of school resources to be higher (e.g., smaller class sizes). So it would make

sense for them to supply the school with extra resources, in the form of their own time

and material resources. As we explore in the next section, this has strong implications for

segregation. But even within public schools, they can choose, if allowed, to do so. This could

explain why parents choose to organize activities in schools, which as Anghel and Cabrales

(2010) document for the case of Spain have a sizable effect on student achievement.

For ease of exposition, it is convenient to define:

θj ≡Ωj

ΩjM

which under assumption (1) implies that

Ωj = θjΩjM = θjυPiM υiMψiM

.

Using equation (18), the equilibrium values for c1i, c2j and ei follow:

c1i =υPiυiψi− 2

3

(υTjυj

γn− Ωj

2

)(19)

c2j =4

3

(υTjυj

γn− Ωj

2

)(20)

ei =υPiυiψi

+1

3

υPiM υiMψiM

(

2υTjυj

υTjM υjM+ θj

)(2υTjM υjM

3γ

)2 ψiMυPiM

− θjω(ω −

(2υTjM υjM

3γ

)2 ψiMυPiM

) (21)

Remark 3 The interaction between parents, schools and education policy generates peer-

effects.

20

The expression for (21) reveals that, in equilibrium, the performance of student i de-

pends on the ability of her peers at different levels. Therefore, the model provides a micro-

foundation for the emergence of peer effects in the classroom without technological assump-

tions.23

First, we obtain peer-group effects in the sense that the student’s own performance

increases in υj. The driving force is the reward scheme at the school level, which depends

on the mean ability of her peers. To our knowledge, this is new in the literature.

Thus, our model predicts that child that attends a school or class with better peers would

improve his effort and educational outcomes. The following are the mechanisms operating in

this situation. First, higher quality students increase school effort. However, parents do react

to the increase in school efforts in the expected direction. Parents of children in schools with

better (worse) peers reduce (increase) their involvement. Under the parametric assumptions

of our model the parental reactions are not strong enough to compensate for the higher

school effort.24 Thus, our model will predict a positive effect for a child that enters a higher

quality school but this will be attenuated by parental behavioral responses. Interestingly,

Cullen, Jacob, and Levitt (2006) and Pop-Eleches and Urquiola (2013) find that the parents

of children that enter elite tracks or schools with better peers spend less time helping their

children with homework than the counterfactual. Pop-Eleches and Urquiola (2013) find

evidence that students benefit academically from access to schools and tracks with better

peers. However, Cullen, Jacob, and Levitt (2006) and Clark (2010) find scant evidence of

improvement in educational outcomes.

Second, performance is affected by the ability of the median student, the characteristics

of her peers (ability and parental involvement) and her teacher’s ability. This sort of peer-

cohort and teacher effects result from the determination of the school resources that affect

classroom motivation in public schools. Overall, we view these results as a cautionary note

regarding empirical analyses which aim at measuring the effect of different education policies

as if they were exogenous to the political process.25

The evidence on the existence of peer effects has been the object of some recent con-

troversy. Some papers, such as Imberman, Kugler, and Sacerdote (2012) using data from a

natural experiment find strong evidence of their presence, whereas other papers, mostly us-

ing evidence from lotteries or entry exams into selective school do not (e.g. Abdulkadiroglu,

23In Appendix A.2, we discuss an extension with standard exogenous peer effects, where the human capitalof the child is multiplied by 1 + λej .

24This is straightforward to check by looking at the derivatives of c1i and c2j with respect to υj in equations19 and 20.

25A related point appears in Besley and Case (2000).

21

Angrist, and Pathak (2014) and Dobbie and Jr. (2011)).26 The most important aspect of this

controversy, from our point of view, is the fact that the peer effects are most likely happening,

as our paper suggests, from a subtle interaction between the peers and school organization.

This can be seen, for example, in the fact that Abdulkadirolu, Angrist, Dynarski, Kane, and

Pathak (2011) pilot schools, which operate like public schools but are more selective produce

no measurable outcomes, whereas charter schools, which change organization at the same

time that students are more strongly selected to present some changes. Similarly, Duflo,

Dupas, and Kremer (2011) find that strong peers have an effect only in some kind of schools,

those that are non-tracking.

As in the case of homogeneous parents and children, an economy-wide increase in parental

opportunity costs induces a reduction of involvement by the parent (the effect of ψi on c1j is

negative) which is partially compensated by the increased involvement of the school system

(the effect of ψiM on c2j is positive). But because of the link between the median child and

individual effort, changes in the distribution of income (or talent) can affect outcomes as

well. For example, an increase in ψiM will generate (for fixed θj) an increase of resources

(a decrease in n) which will have a positive effect on lower-income household even if their

incomes do not change. Thus, a rising tide lifts all votes in this case.27 And the other

way around, if ψiM is unchanged (or almost, again for fixed θj) in an environment where

mean income is increasing markedly, there will be few changes in school outcomes (or even a

regression, because of the negative reaction in c1i of very high income households) at a time

when GDP is increasing.

Equation (21) underlines an important issue regarding the generalization of partial equi-

librium studies of stratification to the entire education system. In general, reduced form

work (e.g., Duflo, Dupas, and Kremer (2011)) will focus on isolating the direct effect of υj

on Hi. However, a generalized system for tracking students affects both υj and υjM . Thus,

the general equilibrium implications may differ from the partial equilibrium ones through

the political determination of the resources in the school system. We explore this issue and

others in the next section.

26Clearly, there is not a consensus yet, but the fact that such schools are so heavily oversubscribed makeus doubt that the effect is nonexistent. It is possible that the schools do something to the students that thestandardized exams fail to pick up, but which later earnings data would show, as Chetty, Friedman, Hilger,Saez, Schanzenbach, and Yagan (2011) have shown for the STAR experiment.

27Corcoran and Evans (2010) find that 12 to 22 percent of the increase in local school spending in the U.S.over the period 1970-2000 is attributable to rising inequality.

22

4 Endogenous segregation

Our model emphasizes the interdependencies established between (parents and school) moti-

vational schemes, class resources and children effort. As a consequence, student performance

depends on group and cohort peer effects. The implications of this finding run deeply into

the different variables of the system. We introduce private schools and investigate sorting in

the school market and the effect of policies inducing segregation, like school vouchers.

Remark 4 Notice that in all equilibrium values parental characteristics always enter as the

ratio ψi/υPi and teacher characteristics as the ratio γ/υTj . Hence, from now on we normalize

and let ψi = ψi/υPi, Ti = T/υPi and γj = γ/υTj .

The differential sensitivity of different types of parents to the composition of classrooms

has segregation-inducing effects that we now analyze. The analysis will gain in clarity if

we identify first a generic condition for an assignment equilibrium with sorting. Once a

condition for this type of equilibrium is identified, we can then verify if it is satisfied for

specific attributes, like talent or income, and a mixed system with public and private schools.

Formally, consider a generic attribute ξ. Take two schools ξ1 and ξ2, where ξ is an average

of that attribute in the school. Parents are allowed to send their children to a private school

paying a fee. The differential willingness to pay of two parents, with types ξi, ξi′ and ξi > ξi′

is:ξ2∫ξ1

∂UPi∂ξj

dξj −ξ2∫ξ1

∂UPi′

∂ξjdξj =

ξ2∫ξ1

(∂UPi∂ξj

−∂UPi′

∂ξj

)dξj

which implies that so long as ∂UPi/∂ξj is monotonic in ξi the willingness to pay is mono-

tonic. This type-monotonicity in relative gains is what leads to an equilibrium with school

segregation by types. More precisely, consider a finite set of schools l ∈ 1, ..., L, each

with nl slots. Assume as well that each nl is high enough so that the compositional impact

of changing one child’s type on the ξl of school l is small. Order arbitrarily the available

schools. We denote by top-down sorting the following assignment of children into schools

according to their type. School 1 gets assigned the n1−highest type children, school 2 the

n2−highest type children among the remaining ones, and so on until all children are assigned

to one (and only one) school. The top-down sorting leads to a segregated school structure

with types stratified from higher to lower. Namely, given two schools l > l′ and two children

i, i′ that are assigned to either school by top-down sorting, then, ξi > ξi′ and ξl ≥ ξl′ . To

ensure that this inequality is strict for at least one pair of players in two different schools,

23

we assume that two successive schools cannot be fully occupied by players of the same type.

To join a school l, parents must pay a fee pl to the owner of the school l. The last school

(or set of schools) in the list is public, free and has enough capacity for N students (the full

group). We say that an assignment of children to schools and a vector of school prices forms

an equilibrium when, given the prices, no individual prefers to change schools and either a

school is full or its associated fee is zero.

Proposition 1 There exists an assignment equilibrium with top-down sorting if whenever

Pi and Pi′ are such that ξi > ξi′ we have that

∂UPi

∂ξj− ∂UPi′

∂ξj> 0 (22)

Letting ξi∗(l) be the type of the lowest type parent in school l, the fee for a full school l is

defined recursively as:

pl =

ξl∫ξl+1

∂UPi∗(l)∂ξj

dξj + pl+1, l = 1, ..., L− 1, (23)

and pL = 0.

Proof Please see Appendix B.

This condition provides a test for the existence of endogenous segregation in different

settings and considering different attributes like income or student talent.

4.1 Segregation with private schools

We explore the emergence of sorting in a framework with private schools. To this end, we need

first to describe the governance structure of these schools. Once private schools’ behavior is

discussed, it has to be shown that this structure satisfies the condition for segregation given

by equation (22).

School behavior Private schools announce fees and allow parents to run schools as clubs.28

More precisely, once school l is formed, the headmaster chooses education policies (in

our case, c2l and nj) to maximize the utility of the median parent. For simplicity we assume

28A similar school governance is assumed, for example, by Ferreyra (2007).

24

in this section that there is heterogeneity in only one dimension υi and ψi = ψj for all i, j.

Parents cover the running costs of the school in addition to paying the entry fee pl. Hence,

after school is formed the headmaster maximizes:

UPiM =

(υiM

ψ+c2l

2

)υiM +

(Ti −

1

2

((υiM

ψ

)2

−(c2l

2

)2))

ψiM

− ω

2n2l

+

(Tl −

1

2nlc2l

(ΩlM +

c2l

2

))γl, (24)

which represents the utility of the median parent in the school l. Notice that the optimal

education policy depends on the characteristics of the median student. This feature differs

from the case of a public school where both the resources determined by the policymaker

and the incentives decided by the headmaster are based on the mean student abilities. The

utility of any parent in school l is:

UPi =

(υi

ψ+c2l

2

)υi+

(Ti −

1

2

((υi

ψ

)2

−(c2l

2

)2))

ψ− ω

2n2l

+

(Tl −

1

2nlc2l

(ΩlM +

c2l

2

))γl,

which can be expressed in terms of UPiM as

UPi = UPiM +1

2

(υ2i

ψ−υ2iM

ψ

)+ (υi − υiM )

c2l

2+

(Ti +

1

2

(c2l

2

)2)(

ψi − ψiM). (25)

At this point, we impose the following assumption for ease of computations:

Proposition 2 Let Pi and Pi′ be such that υi > υi′ , then:

∂UPi∂υl

− ∂UPi′

∂υl> 0 (26)

Proof Please see Appendix B.

Remark 5 This result establishes (22) and hence, by proposition (1), it demonstrates the

existence of an assignment equilibrium with top-down sorting.

In our model, a private school attracting students from the public system affects the

policy variables in a predictable way. Higher student talent induces an increase in school

system resources and the power of school incentives. Students enrolling in a private school

25

(and hence leaving the public school) tend to come from the upper parts of the talent

distribution. From equation (18) one can see that these children leaving the public schools

would entail automatically an increase in n. Similarly from equation (9) one can see that c2j

(school incentives) are directly reduced through the effect of the increase in n.29 This effect

is relevant for evaluating the effect of vouchers, to which we now turn.30

4.1.1 Discussion

The presence of private schools may lead to sorting.31 In line with the literature, we can

discuss the effect of increasing the school choice through vouchers (see e.g. Epple and

Romano (1998), Urquiola and Verhoogen (2009)). Our distinctive feature is our focus on the

endogenous determination of peer-effects, and hence school quality and policy choices, as a

result of the interaction of parents and the school system.32 Taking the interaction between

parents and the school system seriously also has ramifications for estimating the effect of

vouchers on sorting outcomes. 33

Consider the implementation of a voucher scheme subsidizing private schools. The

voucher will induce peers with higher talent to leave the public school.34 As discussed

above, this implies a decline in incentives and resources received by the students who stay

in the public school.35 This result would follow mechanically from assuming the existence

of peer effects as in most of the literature (e.g., Benabou (1993)). In our model, where

peer-effects are endogenously generated by the interaction of parents and the school system,

there is an amplification effect via the reaction of school resources and incentives to changes

in the average ability of the peers. So, the negative effect of sorting on students left behind

is greater than the estimates of a model with exogenous peer-effects would suggest.36

29McMillan (2004) in an otherwise quite different model, also finds that public schools can decrease theirperformance in the presence of vouchers.

30In our model, the authorities invest more in public education the higher its marginal productivity, andthis is why a lower median ability level in the class would decrease public funds. It is conceivable, though,that in the short run, public funds may be fixed. In that case, the fact that some children moved to a privateschool would increase per capita resources in the public one.

31We have described an equilibrium with private schools and sorting. Clearly, there are other equilibriawere parents do not expect sorting, and as a result, it does not happen, and there are no fees due to abilitysorting,.

32Integrating in a analytically tractable framework peer-effects, school quality and education policies, bothin terms of education incentives and resources, is, to the best of our knowledge, novel in the literature.

33In particular, the estimation of computational/structural general equilibrium models have become acommon tool for policymakers to understand the impacts of various educational policies.

34This effect has been empirically uncovered by many studies. See for example, Howell and Peterson(2002) for the case of the US, Hsieh and Urquiola (2006), for Chile or Ladd (2002) for New Zealand.

35Altonji, Huang, and Taber (2004, 2010) provides evidence of this effect for the case of the U.S.36Ferreyra (2007) finds that a generalized voucher scheme even if positive in terms of welfare generates a

26

Also, when students can afford moving to better schools after receiving a voucher, the

positive effect might also be smaller than the pre-voucher situation would suggest. This

is because the receiving school would enroll on average less talented students, and its best

students would like to move to a better private school. The new composition of the private

school would entail a new median student and thus affect the level of resources and school

incentives. Our effect hinges critically on the reactions of the actors involved in the education

process and the consequence of neglecting the implied feedback effects would overstate both

the gains by those favored by the voucher policy and understate the losses suffered by those

who stay in the public school.

5 Concluding remarks

In this paper, we study a model of education where student learning effort and outcomes,

parental and school behavior, and public resources devoted to education are endogenously

determined in an integrated and tractable framework. Our model, provides a rationale

for why the evaluation of educational interventions often provides mixed results. Beyond

rationalizing this phenomenon ex-post, the paper serves as a warning: many evaluation

exercises in education may be seriously compromised by issues of external validity. Indeed,

we show that the effects of changing educational inputs on educational outcomes depend

crucially on the sources of variation that cause this change. For this reason it may be hard

to formulate education policy based on a menu of evaluation results.

The model also provides a microfoundation for peer effects. Groups of children with

higher average ability are more “profitable” to manage by teachers, who as a consequence

exert more effort in them. Then, any child will benefit from their presence in the school.

Peer effects, as in other models, produce an incentive for sorting. We show that in some

circumstances (e.g., when teaching technology favors low variance classrooms) sorting can be

Pareto improving. Even in this case, the welfare gain from sorting is not evenly distributed,

which can explain the ambiguous empirical evidence on sorting. The potentially ambiguous

effect of changes in classroom composition carries implications on different education policies.

In Albornoz, Berlinski, and Cabrales (2010), we extend our model to investigate the effects

of sorting according to talent, cultural preferences or religious beliefs. We show that the

resulting changes in classroom composition may reduce parents’ and school involvement

and resources in disadvantaged classes. As a consequence, the existing evaluations of these

negative effect on poor students.

27

policies may need to be re-evaluated upon this light.

It is clear that there are circumstances when higher ability children are not necessarily

those in which parents want to invest more effort. For example, if the objective of a parent is

to have her child get into an Ivy league school and she is so talented that even without effort

the goal would be achieved, there is no point in making the investment in incentives. On

the other hand, a slightly less talented child may be on the verge of achieving the goal and

some investment in incentives could be indeed profitable. Thus, in reality the relationship

between talent and parental investment may be nonmonotonic. Introducing explicitly these

nonmonotonicities would not change the relationship between parental opportunity costs

and investments, as well as the reaction to this by the education system. Our specific results

on segregation would indeed change. But the most important message is that heterogene-

ity in talent or opportunity cost create incentives for segregation through the responses of

both politicians and educators to the school composition. And this message would remain

unaffected by a potential nonmonotonic relationship between talent and parental investment.

The richness of the model allows it to be used in further research. The political aspects of

school choice, for example, are barely scratched in this paper. Since the political authorities

have a single instrument, school resources, and preferences over this instrument are single-

peaked, we can resort to the median voter theorem in discussing the policymaker’s choice. If

there were more instruments (say, because the level of funding of charter schools is an elec-

toral issue) more challenging (and more interesting) political interactions involving education

could be studied (as in, for example, Boldrin and Montes (2005) or Levy (2005)). Another

aspect we have not explored is that of teacher sorting and teacher peer effects (something

that Jackson (2009), Jackson and Bruegmann (2009), and Pop-Eleches and Urquiola (2013)

have documented). We leave this sort of work for future research.

28

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A Appendix A: Extensions and robustness checks

A crucial insight of this paper is that the empirical analysis on the effect of school resources

should control for changes in parental involvement to derive meaningful estimates. In this

appendix, we discuss variations of some assumptions made throughout the analysis to clarify

whether these are critical to obtain our results.

A.1 Monetary investments in learning

If school performance is determined alone by effort and talent, then it is natural to consider,

as we did so far, that parents induce more effort through time-consuming motivational and

monitoring activities. However, student outcomes may also be enhanced by other activities

with a direct effect on learning, such as paying for private tuition. Dealing with this concern

33

requires changes to the production function of human capital (Hi). Therefore, we assume

now that children performance depends on a function of monetary resources paid by their

parents.37 To study how parental (monetary) education investments interplay with school

(monetary) resources, schools are also allowed to allocate monetary resources to enhance

learning. School (monetary) resources can be used, for example, to hire personal tutors or

teaching assistants. For simplicity, we assume that school monetary resources are chosen by

the policy maker. We also assume that monetary education resources are substitute to time

effort in order to focus on their direct effect. As an additional advantage, this assumption

greatly simplifies the analysis since we can abstract from the determination of student effort.

For a formal analysis, let Hi, UPi and UPM be specified in the following way:38

Hi = (gi + gPM)1/2

UPi = 2 (gi + gPM)1/2 − ηi2g2i

UPM = 2n∑i=1

(gi + gPM)1/2 − ϕ

2g2PM

Where gi gPM represent monetary investment in learning by parents and the school,

respectively. The first order conditions of this problem are:

(gi + gPM)−1/2 = ηigin∑i=1

(gi + gPM)−1/2 = ϕgPM

Interestingly, it readily follows that the policy maker increases gPM as a response of

parents lowering gi, and vice versa. If we take the analysis further, we obtain that in

equilibrium:

37Clearly, monetary resources do not mean that children receive cash for their performance.38For simplicity in this discussion we subsume school decisions with those of the policymaker.

34

g = (1

η)23

(1

nηϕ

+ 1

) 13

gPM =n

nϕ23 + ϕ

η13

An important implication behind these expressions is that the impact evaluation of pub-

lic education expenditure requires controlling for parental investments, even if we consider

that parental monetary involvement as a substitute for learning effort; a similar insight we

obtained when focusing on the effect of motivation on student learning effort.

A.2 Direct peer effects

Peer effects in our model arise so far only because a class with better (worse) students induces

the teacher to make a higher (lower) effort. But in reality it is plausible that students benefit

(or harm) one another directly, and not only through their effect on the effort level of teachers,

or the investment of policymakers. We now also add this kind of direct peer effects, where

the human capital of a child is affected by the average effort in the school. Specifically, let

us assume that

Hi = υiei (1 + λe1j)

This immediately implies, following the same steps as before that

c1i = max

υPiυiψi

(1 + λe1j)−c2j

2, 0

c2j =

υTjυj

γnj(1 + λe1j)−

c1j

2

and hence

c1j =υPiυiψi

(1 + λe1j)−c2j

2

e1j =1

32− υPiυi

ψiλ− υTjυj

γnjλ

(υPiυiψi

+υTjυj

γnj

)

e∗i =υPiυiψi

+υTjυj

γnj+λυPiυiψi

+ λυTjυj

γnj− 1

2

32− υPiυi

ψiλ− υTjυj

γnjλ

(υPiυiψi

+υTjυj

γnj

)

35

As can be readily seen by inspecting the previous expressions, the direct peer effects

make the impact of all the exogenous variables larger, but there are no additional qualitative

effects.

A.3 Parent and teacher complements

For tractability reasons, let us assume that human capital is characterized by Hi = υieiα,

with α < 1.

If parents’ and teachers’ efforts are complements so that

c = c1c2

the optimal student action is

e = c1c2. (27)

Substituting this expression into the parents’ utility we obtain

UP = (c1c2)α υ +

(T − c1 (c1c2)

2υP

)ψ.

The first-order condition for the parents’s problem is then

αcα−11 cα2υ − c1c2

ψ

υP= 0.

αcα2υ − c2−α1 c2

ψ

υP= 0

Given that this condition is sufficient, the optimal choice of the parent is

c1 =

(αυPυ

ψc1−α2

) 12−α

, (28)

which is always non-negative given that producing the rewards requires investing parental

time. By substituting the optimal choice of children’s effort into the utility function of the

headmaster we obtain:

UHM = (c1c2)α υ +

(T − nc2 (c1c2)

2υT

)γ

36

It follows that an interior solution for the headmaster’s optimization problem implies

c2 =

(αυTυ

γnc1−α1

) 12−α

, (29)

From (28) and (29) it is clear that the equilibrium values of c1 and c2 move in opposite

directions even though in this version of the model they are technological complements.

A.4 Endogenous quality of parental involvement

In the core of the paper, we associated productivity of parental motivation with something

innate (exogenous), like talent or capacity of persuasion. However, for a given unit of time

investment (and a given talent), parental motivation may also differ according to different

levels of attention or commitment to the task of motivating student effort. In our model,

this dimension can be captured by treating υPi as an endogenous variable. Consider that

the utility of parents is now defined by:

UPi = υiei +

(T − c1iei

2υPi

)ψi −

1

2υ2Pi, (30)

where υPi resumes the quality of parental effort (attention) and involves quadratic costs.

For concreteness, we abstract from motivation provided by the school system. This implies

that effort is solely determined by c1i. From the first order condition with respect to υPi , we

obtain:

υPi =3

√c2

1i

2ψi

Notice that, for a given c1i, attention increases with the opportunity cost of time. Now,

if we solve for the optimal level of c1i, we obtain:

c1i =υ3i

2ψ2i

The variation explored in this section delivers an interesting new insight: when the qual-

ity of parental involvement is endogenous, the strength of rewards becomes more sensitive

to student talent and (in the opposite direction) to parental opportunity cost of time, al-

though in a structural way attention and the opportunity cost of time are naturally positively

associated.39

39A similar analysis would carried over with similar insights if we dealt with endogenous quality of effort

37

A.5 Other population targets

Headmasters care about the learning outcomes of a whole distribution of students attending

their school. In our main analysis, we assumed that the relevant moment for the headmasters

was the mean of the outcome distribution. This is, of course, a crude description of the

reality and headmaster (as well as policy makers) target different measures of education

achievements. In order to obtain more generality, suppose that the headmaster has a utility

such that he cares about a weighted average of the average student and the n-th order

statistic of the human capital distribution (Hi) at school j, denoted H(n)j . Then, let υ

(n)j be

the talent corresponding to H(n)j and assume

UHMj = λ1

Nj

∑i∈j

Hi + (1− λ)H(n)j +

(T − nj

Nj

∑i∈j

c2j (c1i + c2j)

2υTj

)γ.

Observe that λ ∈ [0, 1] parameterizes the weight given to each goal. Following the

previous analysis, we obtain that the equilibrium individual effort is given by:

ei =υPiυiψi

+1

3

υPiM υiMψiM

(

2υTj

(λυj+(1−λ)υ

(n)j

)υTjM

(λυjM+(1−λ)υ

(n)jM

) + θj

)(2υTjM

(λυjM+(1−λ)υ

(n)jM

)3γ

)2ψiMυPiM

− θjω(ω −

(2υTjM

(λυjM+(1−λ)υ

(n)jM

)3γ

)2ψiMυPiM

)

This expression is similar to 21 since it relates individual student effort to the achieve-

ment of the targeted goal at the school level and at the school attended by the median child.

Importantly, this means that any measure of inequality that is captured by an order statis-

tic may be easily accommodated within our formulation, and the nature of the endogenous

peer-effects are qualitatively unchanged, although they would depend on different features

of the classroom composition. Notice, however, that other measures of human capital in-

equality based on different moments of the distribution would be more difficult to handle

analytically, although this result suggests that it is unlikely they would yield qualitatively

different implications.

for teachers.

38

A.6 Relative performance evaluation as the target

What would happen to school motivation effort if headmasters had an imposed goal to

achieve a certain average student outcome? Let HR be the human capital of average student

in the reference (R) school and assume that:

UHMj = κI 1Nj

∑i∈j(c1i+c2j)υi≥HR

+

(T − nj

Nj

∑i∈j

c2j (c1i + c2j)

2υTj

)γ.

under these circumstances in equilibrium the effort of the c2j is

c2j =HR − 1

Nj

∑i∈j c1iυi

υj

iff

κ ≥ γnjNj

∑i∈j

HR− 1Nj

∑i∈j c1iυi

υj

(c1i +

HR− 1Nj

∑i∈j c1iυi

υj

)2υTj

Since in equilibrium

c1iυi =υPiυ

2i

ψi−HR − 1

Nj

∑k∈j c1kυk

2υjυi

we have (after some algebra) that

1

Nj

∑k∈j

c1kυk =2

3Nj

∑k∈j

υPkυ2k

ψk− 1

3HR

so c2j is positive iff

κ ≥ γnjNj

∑i∈j

23HR− 2

3Nj

∑k∈j

υPkυ2k

ψk

υj

c1i +23HR− 2

3Nj

∑k∈j

υPkυ2k

ψk

υj

2υTj

Notice that the effect of increasing the target is to increase effort for those schools which are

in a position to achieve it, but it decreases the effort to zero for those with low types.

Online Appendix: Proofs

Proof of lemma 1

39

Proof Notice that the preferences of an arbitrary parent i with respect to a class size

level n (once he takes into account the taxes that the policymaker will have to levy to pay

for the costs of such class size) is:

UPi =

(υPiυiψi

+1

3

(2

γnυTjMυj − Ωj

))υi +

+

(T − 1

2υPi

((υPiυiψi

)2

− 1

9

(2

γnυTjυj − Ωj

)2))

ψi −ω

2n2,

so that

sign

(∂UPi∂n

)= sign

(−υi

3

(2υTjυj

γn2

)−

2υTjυj

9υPiγn2

(2

γnυTjυj − Ωj

)ψi +

ω

n3

)= sign

((−

2υiυTjυj

3γ+ Ωj

2υTjυjψi

9υPiγ

)n−

4υ2Tjυ2jψi

9υPiγ2

+ ω

)

and therefore the sign of the derivative of UPi with respect to n can change sign only once.

This means that there is at most one interior critical point. For this critical point to be a

maximum it is sufficient that

−2υiυTjυj

3γ+ Ωj

2υTjυjψi

9υPiγ< 0

Ωj =1

Nj

∑i∈j

υPiυiψi

< 3υPiυiψi

And this is verified if1

Nj

∑i∈j

υPiυiψi

< 3 mini∈1,...,N

υPiυiψi

.

Proof of proposition 1

Proof A parent of a child in school l with type ξi does not want to move the child to

school l + 1 provided that:

UPi(ξl)− pl ≥ UPi

(ξl+1

)− pl+1

UPi(ξl)− UPi

(ξl+1

)≥ pl − pl+1.

40

Such parent will have a type such that ξi∗(l−1) ≥ ξi ≥ ξi∗(l). Then we have that:

UPi(ξl)− UPi

(ξl+1

)=

ξl∫ξl+1

∂UPi∂ξj

dξj

≥ξl∫

ξl+1

∂UPi∗(l)∂ξj

dξj

= pl − pl+1,

where the inequality is true by (22). Similarly a parent of a child in school l with type ξi

does not want to move the child to school l − 1 provided that:

UPi(ξl)− pl ≥ UPi

(ξl−1

)− pl−1

pl−1 − pl ≥ UPi(ξl−1

)− UPi

(ξl)

Remember that ξi∗(l−1) ≥ ξi ≥ ξi∗(l). Thus:

pl−1 − pl =

ξl−1∫ξl

∂UPi∗(l−1)

∂ξjdξj

≥ξl−1∫ξl

∂UPi∂ξj

dξj

= UPi(ξl−1

)− UPi

(ξl),

where, again, the inequality is true by (22).

Proof of proposition 2

Proof

The first order conditions associated with (24) are:

1

2υiM +

c2l

4ψ − γl

2nl(ΩlM + c2l

)c2l = 0,

2ω

n3l

− γl(

ΩlM +c2l

2

)c2l = 0.

41

These conditions imply that:

nl =2υiM + c2lψ

2γl(ΩlM + c2l

)c2l

,

2ω

(2γl(ΩlM + c2l

)c2l

2υiM + c2lψ

)3

− γl(

ΩlM +c2l

2

)c2l = 0.

Using these conditions, by equation (25), we can calculate the following derivatives:

∂UPi∂υiM

=∂UPiM∂υiM

− υiM

ψ− c2l

2+

(υi − υiM )

2

∂c2l∂υiM

(31)

∂c2l∂υiM

= −6ω

(2γl

(υiMψ

+c2l

)c2l

2υiM+c2lψ

)2(−2γlc

22l

(2υiM+c2lψ)2

)

6ω

(2γl

(υiMψ

+c2l

)c2l

2υiM+c2lψ

)24γl

(υ2iMψ

+2c2lυiM

)+2γlc

22lψ

(2υiM+c2lψ)2

− γl (υiψ

+ c2l

) > 0 (32)

Let Pi and Pi′ be such that υi > υi′ then, from (31), we obtain:

∂UPi∂υiM

− ∂UPi′

∂υiM=

(υi − υi′)2

∂c2lM∂υiM

The result then follows from (32).

Appendix C: Algebra (not intended for publication)

n derivation

UPM = (tg)1/2 +2υ2

3

(υPψ

+υTγn

)+

(T − 2υ2

9υP

(2υ2

P

ψ2+υPυTψγn

− υ2T

(γn)2

))ψ − ω

2

1

n2− t

42

−2υ2

3

(υTγn2

)− 2υ2

9υP

(−υPυTψγn2

+ 2υ2T

γ2n3

)ψ + ω

1

n3= 0

−2υ2

3

(υTγ

)− 2υ2

9υP

(−υPυT

ψγ+ 2

υ2T

γ2n

)ψ + ω

1

n= 0

−2υ2

3

(υTγ

)+

2υ2

9

υTγ− 2υ2

9υP

(2υ2T

γ2n

)ψ + ω

1

n= 0

−4υ2

9

(υTγ

)− 2υ2

9υP

(2υ2T

γ2n

)ψ + ω

1

n= 0

n =ω − 2υ2

9υP

(2υ2Tγ2

)ψ

4υ2

9

(υTγ

)2υ2

3

(υPψ

+υTγx

)+

(T − 2υ2

9υP

(2υ2

P

ψ2+υPυTψγx

− υ2T

(γx)2

))ψ − ω

2

1

x2

Gradient is1

x3ω − 2

9υ2 ψ

υP

(2

x3γ2υ2T −

1

x2γψυPυT

)− 2

3x2γυ2υT = 0

Solution is:

14γυ2υP υT

(9γ2ωυP − 4υ2ψυ2T )

if γ 6= 0 ∧ υ 6= 0 ∧ υP 6= 0 ∧ υT 6= 0 ∧ 94γυ2

ωυT− 1

γψυPυT 6= 0

n =ω − 2υ2

9υP

(2υ2Tγ2

)ψ

4υ2

9

(υTγ

)=

ω − ψυP

(2υT υ

3γ

)2

(2υ3

)2 υTγ

Comparative statics homogeneous case

Comparative statics for n

n =1

γ

ω(2υ3γ

)2 − ψ

=

γω(2υ3

)2 −ψ

γ

43

∂n

∂ω=

1

γ

1(2υ3γ

)2

> 0

∂n

∂γ=

ω(2υ3

)2 +ψ

γ2> 0

∂n

∂ψ= −1

γ< 0

∂n

∂υ= −

γω2υ(

23

)2(2υ3

)4 < 0

Comparative statics for c1, c2, and e with respect to υ

c1 =2υ

3ψ

2ω − 3(

2υ3γ

)2

ψ

ω −(

2υ3γ

)2

ψ

=

2

3ψ

2ωυ − 3(

23γ

)2

ψυ3

ω −(

2υ3γ

)2

ψ

44

∂c1

∂υ=

2

3ψ

[2ω − 9

(2υ3γ

)2

ψ

] [ω −

(2υ3γ

)2

ψ

]+

[2ωυ − 3

(2υ3γ

)2

ψυ

](2

3γ

)2

2υψ[ω −

(2υ3γ

)2

ψ

]2

=2

3ψ

[2ω − 9

(2υ3γ

)2

ψ

] [ω −

(2υ3γ

)2

ψ

]+

[4ω − 6

(2υ3γ

)2

ψ

](2υ3γ

)2

ψ[ω −

(2υ3γ

)2

ψ

]2

=2

3ψ

(2υ3γ

)2

ψ

[(4ω − 6

(2υ3γ

)2

ψ

)−(

2ω − 9(

2υ3γ

)2

ψ

)]+ ω

[2ω − 9

(2υ3γ

)2

ψ

][ω −

(2υ3γ

)2

ψ

]2

=2

3ψ

(2υ3γ

)2

ψ

[2ω + 3

(2υ3γ

)2

ψ

]+ ω

[2ω − 9

(2υ3γ

)2

ψ

][ω −

(2υ3γ

)2

ψ

]2

=2

3ψ

3(

2υ3γ

)4

ψ2 + ω

[2ω − 7

(2υ3γ

)2

ψ

][ω −

(2υ3γ

)2

ψ

]2

and c2 > 0 implies that:

2ω < 6

(2υ

3γ

)2

ψ

and hence

2ω < 6

(2υ

3γ

)2

ψ +

(2υ

3γ

)2

ψ

which leaves ∂c1∂υ

> 0.

c2 =2υ

3ψ

3(

2υ3γ

)2

ψ − ω

ω −(

2υ3γ

)2

ψ

=

2

3ψ

3(

23γ

)2

ψυ3 − ωυ

ω −(

2υ3γ

)2

ψ

45

∂c2

∂υ=

2

3ψ

[9(

23γ

)2

ψυ2 − ω] [ω −

(2υ3γ

)2

ψ

]+

[3(

23γ

)2

ψυ3 − ωυ](

23γ

)2

2υψ[ω −

(2υ3γ

)2

ψ

]2

=2

3ψ

[9(

2υ3γ

)2

ψ − ω] [ω −

(2υ3γ

)2

ψ

]+

[3(

2υ3γ

)2

ψ − ω](

2υ3γ

)2

2ψ[ω −

(2υ3γ

)2

ψ

]2

but c1 > 0 implies

ω >

(2υ

3γ

)2

ψ

and c2 > 0 implies

3

(2υ

3γ

)2

ψ > ω

and therefore that

9

(2υ

3γ

)2

ψ > ω.

Then, ∂c2∂υ

> 0.

e =

2υω3ψ

ω −(

2υ3γ

)2

ψ

∂e

∂υ=

2ω3ψ

[ω −

(2υ3γ

)2

ψ

]+ 2

(2υ3γ

)2

ψ 2ω3ψ[

ω −(

2υ3γ

)2

ψ

]2

=

2ω3ψ

[ω +

(2υ3γ

)2

ψ

][ω −

(2υ3γ

)2

ψ

]2 > 0.

46

Comparative statics for c1, c2, and e with respect to ψ

c1 =2υ

3ψ

2ω − 3(

2υ3γ

)2

ψ

ω −(

2υ3γ

)2

ψ

=

2υ

3

2ω − 3(

2υ3γ

)2

ψ

ψω −(

2υ3γ

)2

ψ2

∂c1

∂ψ=

2υ

3

−3(

2υ3γ

)2[ψω −

(2υ3γ

)2

ψ2

]−[ω − 2

(2υ3γ

)2

ψ

] [2ω − 3

(2υ3γ

)2

ψ

][ψω −

(2υ3γ

)2

ψ2

]2

=2υ

3

−3ψ(

2υ3γ

)2[ω −

(2υ3γ

)2

ψ

]−[ω − 2

(2υ3γ

)2

ψ

] [2ω − 3

(2υ3γ

)2

ψ

][ψω −

(2υ3γ

)2

ψ2

]2

=2υ

3

−ψ(

2υ3γ

)2[3ω − 3

(2υ3γ

)2

ψ

]−[ω −

(2υ3γ

)2

ψ

] [2ω − 3

(2υ3γ

)2

ψ

]+ ψ

(2υ3γ

)2[2ω − 3

(2υ3γ

)2

ψ

][ψω −

(2υ3γ

)2

ψ2

]2

=2υ

3

−[ω −

(2υ3γ

)2

ψ

] [2ω − 3

(2υ3γ

)2

ψ

]− ωψ

(2υ3γ

)2

[ψω −

(2υ3γ

)2

ψ2

]2 < 0

c2 =2υ

3ψ

3(

2υ3γ

)2

ψ − ω

ω −(

2υ3γ

)2

ψ

=

2υ

3

3(

2υ3γ

)2

ψ − ω

ψω −(

2υ3γ

)2

ψ2

47

∂c2

∂ψ=

2υ

3

3ψ(

2υ3γ

)2[ω −

(2υ3γ

)2

ψ

]−[ω − 2

(2υ3γ

)2

ψ

] [3(

2υ3γ

)2

ψ − ω]

[ψω −

(2υ3γ

)2

ψ2

]2

=2υ

3

3ψ(

2υ3γ

)2[ω −

(2υ3γ

)2

ψ

]− 3ψ

(2υ3γ

)2[ω − 2

(2υ3γ

)2

ψ

]+ ω

[ω − 2

(2υ3γ

)2

ψ

][ψω −

(2υ3γ

)2

ψ2

]2

=2υ

3

3ψ(

2υ3γ

)2 (2υ3γ

)2

ψ + ω

[ω − 2

(2υ3γ

)2

ψ

][ψω −

(2υ3γ

)2

ψ2

]2

=2υ

3

3ψ(

2υ3γ

)2 (2υ3γ

)2

ψ − ω(

2υ3γ

)2

ψ + ω

[ω −

(2υ3γ

)2

ψ

][ψω −

(2υ3γ

)2

ψ2

]2

=2υ

3

(3ψ(

2υ3γ

)2

− ω)(

2υ3γ

)2

ψ + ω

[ω −

(2υ3γ

)2

ψ

][ψω −

(2υ3γ

)2

ψ2

]2

but c1 > 0 implies

ω >

(2υ

3γ

)2

ψ

and c2 > 0 implies

3

(2υ

3γ

)2

ψ > ω

thus

∂c2

∂ψ=

2υ

3

(3ψ(

2υ3γ

)2

− ω)(

2υ3γ

)2

ψ + ω

[ω −

(2υ3γ

)2

ψ

][ψω −

(2υ3γ

)2

ψ2

]2 > 0

e =

2υω3ψ

ω −(

2υ3γ

)2

ψ

48

e =2υω

3ωψ − 3(

2υ3γ

)2

ψ2

∂e

∂ψ=

−[ω − 2

(2υ3γ

)2

ψ

]6υω[

3ωψ − 3(

2υ3γ

)2

ψ2

]2 S 0

The effect to is negative if ω > 2(

2υ3γ

)2

ψ.

Comparative statics for c1, c2, and e with respect to $

c1 =2υ

3ψ

2ω − 3(

2υ3γ

)2

ψ

ω −(

2υ3γ

)2

ψ

∂c1

∂ω=

2υ

3ψ

2

[ω −

(2υ3γ

)2

ψ

]−[2ω − 3

(2υ3γ

)2

ψ

][ω −

(2υ3γ

)2

ψ

]2

=2υ

3ψ

(2υ3γ

)2

ψ[ω −

(2υ3γ

)2

ψ

]2 > 0

c2 =2υ

3ψ

3(

2υ3γ

)2

ψ − ω

ω −(

2υ3γ

)2

ψ

∂c2

∂ω=

2υ

3ψ

−[ω −

(2υ3γ

)2

ψ

]−[3(

2υ3γ

)2

ψ − ω]

[ω −

(2υ3γ

)2

ψ

]2

= − 2υ

3ψ

2(

2υ3γ

)2

ψ[ω −

(2υ3γ

)2

ψ

]2 < 0

49

e =

2υω3ψ

ω −(

2υ3γ

)2

ψ

∂e

∂ω=

2υ3ψ

[ω −

(2υ3γ

)2

ψ

]− 2υω

3ψ[ω −

(2υ3γ

)2

ψ

]2

=−(

2υ3γ

)2

ψ[ω −

(2υ3γ

)2

ψ

]2 < 0

50